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Full text of "On the turbulent friction layer for rising pressure"

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NATIONAL ADVISORY COMMITTEE 
FOR AERONAUTICS 

TECHNICAL MEMORANDUM 1314 



ON THE TURBULENT FRICTION LAYER FOR RISING PRESSURE 
By K. Wieghardt and W. TiUmaim 



Translation of ZWB Untersuchungen und Mitteilungen 
Nr. 6617, November 20, 1944 



NACA 

Washington 
October 1951 



..NTSDEPAR-mEMT 
;,ARSTON SCIENCE UBRARY 

RO. BOX 117011 QiiuS>. 

GAINESVILLE. FL 3^t>i 



^VS 



iC i^^HO ^iiir'ir 



NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 



TECHNICAL MEMORANDUM 131^ 



ON THE TURBULENT FRICTION LAYER FOR RISING PRESSURE* 
By K. Wieghardt and W. TilLmann 



Abstract: As a supplement to the UM report 6603, measurements in tur- 
bulent friction layers along a flat plate with 'rising pres- 
sure are further evaluated. The investigation was performed 
on behalf of the Aerodynamischen Versuchsanstalt Gottingen. 

Outline: 1. SYMBOLS 

2. INTRODUCTION 

3. TEST SETUP 
k. TEST RESULTS 

5. ON THE GRUSCHWITZ CALCULATION METHOD 

6. ON AN ENERGY THEOREM FOR FRICTION LAYERS 

7. SUMMARY 

8. REFERENCES 



1. SYMBOLS 

X position rearward from leading edge of the plate 

y distance from wall 

u velocity component in x-direction 

V velocity component in y-direction 
U^V velocity components outside of the friction layer 
p density 
\i viscosity \, of the air- 

V kinematic viscosity 
p static pressure 

"Zur tvirbulenten Reibungsschicht bei Druckanstieg." Zentrale fiir 

wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluft- 
zeugmeisters (ZWB), Berlin-Adlershof, Untersuchungen und Mitteilungen 
Nr. 6617, Kaiser Wilhelm-Institut fiir Stromungsforschung, Gottingen, 
November 20, 19iiU. 



NACA TM ISl^^ 



P 2 
q = 2U 



= p + q 



dynamic pressure 
total pressure 



D ? 
Q = ^ U dynamic pressure outside of the friction layer 



Ux/v 
U62/V 

& 



Reynolds number 

Reynolds number of the friction layer 

friction layer thickness 



6-, = / (1 - 77)^ displacement thickness 



^0 

^6 



62 = 



^3 = 



u 



momentum loss thickness 



dy energy loss thickness 



%2 - ^l/S^ 

Hop — Oo/Oq 



^ form parameters of the velocity profile 



T) = 1 - 



"u(S2)" 

U 



^•f' 



= T 



turbulent shearing stress 
wall shearing stress 

/p p 

j/p U"^ local friction drag coefficient 

mixing length 

2. INTRODUCTION 



For the calculation of laminar boundary layers in two-dimensional 
incompressible flow, numerous calculation methods have been developed 



NACA TM 131^ 



on the basis of Prandtl's boundary -layer equation; in contrast, only the 
semiempirical method of E. Gruschwitz (reference l) and its improvements 
by A. Kehl (reference 2) and A. Walz (references 3^ ^, and 5) are avail- 
able for turbulent friction layers. The reason, as is well known, lies 
in the lack of a mathematical law for the apparent shearing stress t 
which originates by the turbulent mixing of momentum. Prandtl's expres- 
sion for the mixing length so far has led to success only in cases where 
a sufficient niJinber of correct (and also sufficiently simple) data on the 
variation of the mixing length can be obtained directly from the geometry 
of the flow as for instance in the case of the free jet. 'As a basis for 
the Gruschwitz method there serve, therefore, besides the momentum equa- 
tion for friction layers, three statements which are partly empirical 
and lack a theoretical basis. They are as follows: 

A. The velocity profiles in the turbulent friction layer for vari- 
able outside pressure form, after having been made adequately dimension- 
less, a single-parameter family of ciirves, if one disregards the laminar 
sublayer; thus, every profile may be characterized by a single 
quantity (t^) . 

B. A differential equation, likewise derived solely from experi- 
ments, concerning the variation of this parameter in flow direction as 
a function of the pressure variation and of the momentum thickness. 

C. An assumption concerning the wall shearing stress. Gruschwitz 
inserts a constant as first approximation. 

A. Kehl (reference 2) then improved the Gruschwitz method. According 
to his measurements which extended over a larger Re-number range than 
those of Gruschwitz, it is necessary to insert in statement B, for a 
higher Re-number of the friction layer, a function of the Re-number 
U52/V instead of a certain constant b. Furthermore, Kehl obtained 

better agreement between the calculation and his test results by sub- 
stituting in scatement C for the wall shearing stress the value which 
results for the respective Re(52) number at a flat plate with constant 
outside pressure. Finally, A. Walz (references 3^ ^) and 5) greatly 
simplified the integration method mathematically so that for prescribed 
variation of the velocity outside of the friction layer U(x) the mom- 
entum thickness, the form parameter t\, and hence the point of separ- 
ation can be calculated very quickly. No statement is obtained regarding 
the wall shearing stress, since it had, on the contrary, been necessary 
to make the assumption C concerning Tq in order to set up the cal- 
culation method at all. 

Thus it seemed desirable to investigate the friction drag of a 
smooth plate for variable outside pressure. On one hand, direct interest 
in this exists in view of the wing drag; on the other hand, one could 



NACA TM 131^ 



expect an Improvement in the above calculation method if an accurate 
statement regarding Tq could be substituted for the assumption C. In 
order to arrive at the simplest possible laws, friction layers along a 
flat smooth plate were investigated where a systematically increasing 
pressure was produced by an opposing plate. By analogy with the behavior 
of laminar boundary layers the wall shearing stress was expected to 
decrease in the flow direction up until separation, more strongly than 
In case of constant outside pressure. Instead, Tq increased, after a 

certain starting distance, more or less suddenly to a multiple of the 
initial value. A brief report on this striking behavior of the friction 
drag has already been published (reference 6) . In the present report, 
these tests are further evaluated and compared with those of Gruschwitz 
and Kehl. Considerable deviations result in places; however, it was not 
possible to develop a better calculation method with this new test mate- 
rial either. 

Since the application of an energy theorem had proved expedient for 
the calc\ilation of laminar boundary layers (reference 'j) , a theoretical 
attempt in this respect was made for turbulent friction layers, too; 
however, it did not meet with the same success. Merely an interpretation 
for the statement B can be obtained in this manner, which is. however, 
not cogent. 



3. TEST SETUP 



The test setup and program have been described in the preliminary 
report. A new measuring method was developed where with the aid of a 
pressure rake and of a multiple manometer (reference 8) turbulent friction 
layers co\ild be measured quickly and accurately and the computational 
evaluation greatly simplified. 

Friction layers were measured at p = for two different veloc- 
ities U = const., four cases, I to IV, with rising, and one, V, with 
diminishing pressure (figs. 1 to 6) . In cases I and II, p increases 
almost in the entire measuring range linearly with rearward position 
with respect to the leading edge of the plate x, whereas in cases III 
and rv the outer velocity U (U ~ x"^) decreases with a power of x 
(figs. 1 to 6) . 

The velocities were about 20 to 60 meters per second; the test 
section length was 5 meters so that Re -numbers up to 107 were attained. 
The Re-number of the friction layer (formed with the momentum thickness) 
increased to from 2 to 7 X lo5. 



MCA TM IBlij- 

h. TEST RESTTLTS 



Mainly the variation of the wall shearing stress had heen described 
in the brief preliminary report (reference 6); later the measurements 
were evaluated more thoroughly. First, we plotted figures 1 to 6 for 
the different pressure variations: the outer velocity U, the local 
friction drag coefficient c^', the displacemenx thickness 6^, the 

momentum thickness 82^ and the energy loss thickness 5^ .treated in 

section 6; in analogy to the momentum loss thickness 82^ 80 is a 

measure of how much kinetic energy of the flow is lost mechanically due 
to the friction layer, that is, is converted to heat by the effect of 
the friction forces. 

In case of constant outside pressure Cf' depends, for a smooth 
plate, only on the Re-number. The test points from figure 1 lie between 
the formulas for cf' according to L. Prandtl (reference 9)^ F. Schultz- 
Grunow (reference 10 ), and J. Nikuradse (reference 11) which in the test 

range Ux/v = 3 X lO-' to 10' differ only slightly. For rising pressure, 
a slight decrease of c^' with rearward position results at first, 

according to figures 2 to 5j however, after a certain distance c^ ' 
increases more or less suddenly to a multiple of its original amount and 
decreases again only at the end of the test section where for test tech- 
nical reasons the pressure increase could no longer be maintained. 
W. Mangier (reference 12) found the same unexpected behavior in further 
evaluating the measurements of A. Kehl (reference 2). In contrast, c^' 
varies only slightly in case of decreasing pressure. Thus, on one hand, 

-T*- > must be responsible for the strong increase in wall shearing 

stress; furthermore, the Re-number of the friction layer is of importance 
since first a certain starting distance is required. Therefore, we 

plotted in figure 7 c^' against the dimensionless -r- — ^. Our measure- 

■'■ y dx 

ments resulted in a comparatively narrow bundle of curves; however, the 
results according to Kehl -Mangier, drawn in in dashed lines, cannot be 
brought under a common denominator in this manner. The test arrangement 
of Kehl was more general insofar as he had at first a piece of laminar 
boundary layer and, moreover, in some cases first a pressure drop, and 
then an adjoining pressure rise; in our tests, in contrast, the friction 
layer, starting from the leading edge of the plate, had been made tur- 
bulent by a trip wire and the pressure increased monotonically. 



NACA TM 13114- 



In want of a better criterion, we can see from figure 7 that the 
strong increase in friction drag is not to be expected as long as 



^ ^ < 2 X 10-3 
Q dx 

The displacement and momentum thickness as well as the wall shearing 
stress represent only a summary of the development of a friction layer. 
Therefore, we shall consider below the velocity profiles and shearing 

stress profiles. The variation of — against the rearward position 
from the leading edge for constant wall distance is particularly illus- 
trative (figs. 8 and 9)- In case II (fig. 8), — suddenly drops steeply 

in the layers near the wall whereas it decreases continuously in case IV. 
Accordingly, the characteristic lengths S]_, 62, and 60 increase at 

these points more strongly than before. Since the drag coefficient Cf' 

depends essentially on the variation of ^— , one recognizes at once in 

ox 

case II the point of maximum wall shearing stress at x w 3'3 meters. 

The shearing stresses are obtained from Prandtl's boundary- layer 
equation which may be transformed with the aid of the continuity equation 

and of the Bernoulli equation = -UU„ valid outside of the friction 

p dx ^ 

layer. With U^ = -p , % = v^, Uy = ^, and Q = ^ U , one obtains 



S_i_^,li^x_!^ r^^ay (1) 

Sy 2Q U U U U Jq U 



The integration which is still to be performed yields additionally a 
control value for the wall shearing stress 

— = = - / dy (2) 

2Q 2 Jq ^y 2Q 



However, the value for Tq obtained in this manner is not as reliable 
as the one calculated from the momentum theorem because here graphical 
differentiation is applied more often. According to a suggestion by 



MCA TM 131^ 



Professor Betz, the momentum theorem may be transformed for the calcvil- 
atlon of Tq in the following manner (compare reference 6): 



2Q y2+Hi2 dx\ 



-.,) . (h,, - H,,)i 3^ a, (3) 



where a suitable mean value of E-^Q ^ ^l/^2 ^^ substituted for H2_2- 
Then the second term is small compared to the first and plays only the 
role of a correction term so that essentially only one graphical differ- 
entiation has to be performed. 

First, the shearing stress profiles are plotted for constant outside 
pressure in figure 10. The shearing stress t with the appertaining 
wall shearing stress t^ and the wall distance y with the momentum 

thickness S2 are made diraensionless. These dimensionless profiles are 

almost completely identical although a systematic variation with the 

Re-number is recognizable. The profiles t against y for the two 

cases II and IV with press\ire rise follow in figures 11 and 13; 

figures 12 and 1^4- show the corresponding dimensionless shearing stress 

profiles t/tq against y/52- In the case II where the pressure p 

increases approximately linearly, the t -profiles differ considerably 

for various rearward positions, especially the wall distance at which 

T reaches its maximum value is subject to great changes. In contrast, 

-0 2T 
the dimensionless profiles in case IV where U <*< 53(x/l°') ' meters 

per second is valid are very similar. Only the one profile at 

X = 1.99 meters stands out sharply without any perceptible reason. 

According to D. R. Hartree (reference 13) ^ similar velocity and hence 

also shearing stress profiles result for laminar boundary layer in the 

case U ~ x~ since for laminar flow t is simply t = U ^. In the 

turbulent friction layer, the velocity and hence also the shearing stress 
profiles thus vary in this case; however, the latter at least can be 
described - although somewhat forcibly - by a single -parameter family of 
curves, according to figure ik. 

A. Buri (reference 1^) attempted to interpret the shearing stress 
profiles in general as single -parameter family of curves. Bvurl selected 
for the parameter Tg^ the computed wall tangent of the dimensionless 

S2/St' 



shearing stress profile ?„ = — \^—] because for this quantity a 

^o\ay/y^o 

relation may be read off immediately from the boiindary-layer equation. 



8 NACA TM 131^ 

For y=0, u=v=0 and therewith r„ = ^ which yields 

^ To dx -^ 

-— I = -^. The tangent to the shearing stress profile thus calcu- 

^y/y=o ^"^ 

lated is known in general to fit very hadly the shearing stress variation 
in the proximity of the wall determined from measurements. In the tur- 
bulent friction layer, the velocity u decreases only in immediate wall 
proximity - in the so-called lajninar sublayer - to zero so that the tan- 
gent direction deviates from the above calculated value even for very 
small wall distances. This is shown also in figure 15, which represents 
the shearing stress profiles in wall proximity and the pertaining wall 
tangents found by calculation; for constant outside pressure, too, the 
curves against y apparently begin to drop from the point y = out- 
ward with a definite angle rather than continue with a horizontal 
tangent (fig. 10). Nevertheless, Fg^ could be used at first as a com- 
puted quantity for the characterization of a definite shearing stress 
profile. However, in case II, for instance, the same shearing stress 
profile would have to be present for x = 3-19 meters and for 
X = 3-79 meters according to figure 15, which is obviously not true 
according to figure 12. In general, the shearing stress profiles could 
not be characterized by one other parameter alone either. At least two 
quantities would be required for this, for instance, the magnitude and 
the dimensionless wall distance of the maximum ^jj^^^/tq. 

Finally, we calculated from the shearing stress the mixing lengths 

5u 



according to Pranatl's expression t = p2 -— 

oy 



and plotted them in 



Sy 

figures I6 to 19 against the wall distance y or in dimensionless form 
Z/62 against y/52. Like the thickness of the friction layer, 2 

increases more and more with further rearward movement; in case IV, 2 
even increases on approaching the end of the test section to 30 milli- 
meters. The diminishing of the mixing length for large wall distances 
cannot be specified, as is well known, since 2 there is computed as 

the quotient of two small quantities, namely, t and r— • Here 

again I increases linearly in wall proximity. In the tube, the result 
had been 2 = O.^J-y and at the plate for constant pressure, according 
to Schult^ -Grunow (reference 10) 2 = 0.i4-3y (dashed in figs. I7 and 19)- 
Here, at rising pressure, I increases at first again linearly but far 
more rapidly. In case II, 2 attains the maximum value 2 = l.ly and 
in case IV even 2 = 2.0y. Although no fixed relation exists between 
2/y in wall proximity and the wall shearing stress, it is striking in 
figures I7 and 19 that 2/y is largest just for those rearward positions 
where the wall shearing stress Tq also (compare figs. 3 and 5) attains 
its maximum value. 



NACA TM 131^ 



"i. ON THE GRUSCHWITZ CALCULATION METHOD 



Our test material enables us to examine the basis of the Gruschwitz 
method enumerated under section 1. The assumption A according to which 
all turbulent velocity profiles concerned form a single -parameter family 
of curves is again confirmed in figure 20. Here a few velocity profiles 
of the different cases I to V are plotted for which the evaluation 
accidentally had resulted in exactly the same ratio H2^2 ~ ^1/^2^' ^^^ 
profiles u/U against y/Sg with equal H12 ^^^ hence equal t\ 
(compare fig. 27) are in agreement even for different velocities^ rear- 
ward positions^ and pressure variations, thus also after different 
"previous history." This single -parameter quality covers, however, only 
the "visible" turbulent part of the profile; the velocities in the 
laminar sublayer may have a different distribution even for equal t]; 
otherwise, a unique connection would necessarily exist between the wall 
tangent Fg^ and i^ or Hj_2 which is certainly not the case according 

to what was said above . 

The change of the velocity profiles characterized by the param- 
eters r| or H2^2 with change in the rearward position is represented 

in figures 21 and 22 for all measuring series. The weak but throughout 
systematic dependence of the profile on the Re -number for constant out- 
side pressure is remarkable; Niku_adse (reference 11), on the other 
hand, obtained here always the same profile with t] = 0.515 and 
H;L2 = 1-302. 

In order to apply the momentum theorem for the Gruschwitz method, 
the relation between H]_2 and t), necessarily unique for a single- 
parameter profile class, must be known. The results for all profiles 
of our measuring series are plotted in figure 23- All of them lie below 
the curve indicated by Gruschwitz but the majority of Kehl's points also 
lie below the original Gruschwitz curve. Since, however, all the results 
do not greatly deviate from one another, the assumption A may be regarded 
as correct in good approximation. The long dashed curve was calculated 
by Pretsch (reference I5) for power profiles; the power pertaining to 
a certain H-L2 1^ given in the figure at the right. Concerning the 
short dashed curve, compare section 6. 

Masters are different for assumption B. Gruschwitz had obtained 
from his test evaluation 



&P dg(6p) o -, 

— — —^-^ == ari - b, with a = 8.9^+ x lO'-^ and b = h.6l x 10"^ 
Q dx 



10 NACA TM I31U 

8(^2) ^^ "^^^ total pressure at the wall distance 52: 

2 



6(52) = P + |[^(52)] 



Because of tj = 1 - u(52) /U and the Bernoulli equation outside of the 
friction layer p + Q = constant, we may also write 

dg(62) dp -, d(Qii) 

= — P + Q(l - il) = 

dx dx"- -■ dx 



Kehl, who investigated a larger Re-number range, found b to be addi- 
tionally dependent on U52/v; therefore, he plots 

52 dg(S2) 52 d(QTi) 

b = ari ^^ = a.T\ + {h) 

' Q dx ' Q dx 



against U52/V • This has Deen done for our measurements in figure 2h. 

In this diagram, Gruschwitz obtained a horizontal straight lihe 

_-> 
b = constant = ii.6l X 10 and Kehl the plotted curve which, starting 

o 

from U52/V = 2 X 10 slowly drops. These two lines have been drawn 

solidly as far as measurements existed. Our measurements show that at 
least for the cases I and II - that is, for linear steep pressure rise - 
b can be assumed neither as constajit nor as a unique function of U62/V 

since we obtain in this diagram two essentially deviating cuj.ves. The 
same result is obtained also in th^ cases III and IV for higher Re-numbers. 

Only for U62/V < 10 , the deviations of the measuring points may be 
interpreted as scatter. It is not the Kehl relation but the simpler 
Gruschwitz relation which is confirmed here. For U52/V < 2 X 10^ Kehl 

took the drawn variation of b in order to obtain significant calculation 
results directly behind the transition point from laminar to turbulent 

flow. The two points with U62/V < 10^ m the case V are not an argu- 
ment against this b-variation for the reason that the friction layer had 
been made artifically t\irbulent to start with by means of the trip wire. 
Below, we shall attempt a theoretical interpretation of the relation B 
which so far has been set up and investigated in a purely empirical 
manner . 



MCA TM 131^ 



11 



Since the variation of the wall shearing stress proved to be very 
complicated, according to figiires 2 to 6, we cannot improve upon assxjmp- 
tion C which refers to the wall shearing stress. In order to estimate 
the effect of this assumption on the calculation, the cases II and IV 
were calculated with the aid of Walz ' s simplified integration method 
under different assumptions regarding c^' . Figure 25 shows the result 

for the momentum thickness 52- In the momentum theorem, we may put 
^12 ~ constant since It appears only in one term 2 + H]_2 so that 
even great variations of H^^g ^^^ °^ relatively small importance. In 

case II, one obtains better agreement (at least up to x = 3^) between 
calculation and tests by using the expression for c^ ' obtained as a 

function of U52/V for the plate without pressure rise than by putting 
c^' = constant. Conversely, the agreement for case IV is better with 
the assumption c^' = constant. In view of the actual variation of 
Cf ' which varies from case to case, no generally valid rule can there- 
fore be set up for it, even before the region of the steep rise of Cf ' . 

The results of the r\ calculation are represented in figure 26. 
The three different calculation methods worked out by Walz are based 
on the following assumptions: 



Gruschwitz-Walz b = constant = k.6l X 10"3 

and 
c„' = constant = k x 10"3 



Gruschwitz-Walz b = constant = ij-.6l X 10"^ 

and 
Cf.' = 0.025l(U52/v)"^A 



Gruschwitz-Kehl-Walz b = 



0.01614- 



0.85 



log (U&2/V) U52/V - 300 
and 
= 0.0251(U52/V)"^/^ 



12 



NACA TM I31U 



As was to be expected according to figure 2U, the last, most complicated 
method is precisely the one that shows the greatest deviations compared 
to the experiment. In our cases II and IV, the first and simplest cal- 
culation method is the best. The agreement between that calculation and 
the test is still good even in the region of steep rise Cf ' . From there 

on, however, large differences result. 



6. ON AN ENERGY THEOREM FOR FRICTION LAYERS 



Like the approximation methods for laminar boundary layers, the 
Gruschwitz method is based on Karman's momentum equation which is obtained 
by integration of Prandtl's boundary-layer equation with respect to the 
wall distance y. One obtains thereby a statement on the total momentum 
loss of the friction layer. In analogy, a statement on xhe total energy 
loss in the friction layer caused by the friction can be obtained if the 
boundary -layer equation 



pUUjj + OVUy = -Pj^ + 



Ty = PUU^ + Ty 



(5) 



pu. 



(after addition of -p-(ux + Vy) = because of continuity) is first 
multiplied by u and then integrated with respect to y 



i. J%„(| „. . 1 ,2) ,, 



-.8 



pu^v 



+ P ^ / ^x dy = 



UTy dy 



and because of 



2 
pu V 



I U2v = -H U2 rS 



I "Y' "- '' 



J ^v ^y = / ^"^ ^^ = kf p^(l "^ - 1 ^^) ^y 



(6) 



This equation signifies that the loss in kinetic energy per unit length 
transverse to the flow direction in the friction layer equals the rate 
of doing work of the turbulent shearing stresses. 



NACA TM 131^ 13 



For the laminar boundary layer t = [lUy is valid so that 

/ "^Uy dy = i-t / Uy2 dy here signifies the dissipation, that is, the 

energy converted to heat per unit time. 

In turbulent friction layers, in contrast, no simple relation between 
the shearing stress and the velocity profile exists; if it did, the essen- 
tially single -parameter family of tiirbulent velocity profiles woiild have 
to include also a single -parameter profile class of the shearing stresses 
which is not the case according to the test evaluation. If we define, 
corresponding to the momentum loss thickness, an energy loss thickness 



(7) 



^0 
and the following dimensionless 




e = 



work of the shearing stresses _ j _r_ S_ u , o\ 

work against the wall shearing stress Jq t^ Sy U 



we can write equation (6) also as 



e = 



1 



c^'u3 



I^K) (9) 



Since we have thus obtained one new equation with two new unknowns, 
this energy theorem at first does not help us any further either. How- 
ever, we can calculate the energy loss thickness 5o from the velocity 

distributions and obtain, due to zhe single -parameter quality of the 
velocity profiles, a fixed relation between the quantities &]_, 62, and 

6^. Plotting thus the ratio H02 = B3/S2 against H-]_2 = ^±/^2 ^°''^ ^-^^ 
profiles measured, we obtain figure 27. All points come to lie, with 
only very small deviations, on one "street." For power profiles 

^/U = (y/S)"^ the long dashed curve 

AHng 



Ik NACA TM 1314 

vould result with A = U/B and B = l/3- A good fairing curve for the 
actual profiles at pressure rise and di op results if we put A = I.269 
and B = 0.379' On the basis of this empirical relation, 63 may now 

be calcxilated from 5]_ and 62 with sufficient accuracy. 

If one more relation could be found for the ratio e as well, it 
would be possible to calculate from the new equation (9) for instance 
cj.' . The calculation of e from the measiiring results is made difficult 

by the fact that it results as the quotient of two quajitities which can 
both be fo\ind only by graphic differentiation. This explains the great 
variation of the e-values plotted for the different cases in figure 28. 
On the whole, e varies only comparatively little; even when c^.' for 

instajice in case IV increases from 3 "to 16 X 10"^, e remaint between 
0.8 and 1.2. At first glance, e seems to have, according to the 
defining equation (8), the significance of an efficiency which could 
not exceed 1. Actually, however, e may assume any arbitrary value, 
according to the velocity and shearing stress profile concerned; in the 
immediate proximity of tiirb^olent separation, above all, where Tq 
vanishes, e may assume arbitrary magnitude. 

Because of the inaccuracy of the calculation from the test data, it 
was not possible to determine for e a relation to the other boundary- 
layer quantities. However, one can set up an interesting analogy between 
the energy equation and Gruschwitz ' s assumption B. If one substitutes 
the above relation for H02 which was found experimentally in the energy 

d52 
equation and eliminates with the aid of the momentum theorem, one 



dx 



obtains 



U„ -R dH-|p Htp(A - 2e) + 2eB 

— + ^ = J^ c^' (11) 

U Hi2(Hi2 - B)(Hi2 " 1) dx 2AHi2(Hi2 " 1) ^ 



On the other hand, one may differentiate out the Gruschwitz -Kehl relation 
(assumption B), equation (k) , and v—ite 



^ _!_ J^ di] dHi2 ^ b - aT] 



(12) 
U 2ti dH]_2 dx 2ii 



regarding t) as a function of H-]^2- ^ = ^(^12)- We now assume that 
the relation B follows from the energy theorem. If this is the case. 



MCA TM I31U 15 



the left sides of equations (ll) and (12) must, first of all, be iden- 
tical. We therefore equate, as an experiment, the left sides of the 
equations and obtain a differential equation for t\(^E±2) > "the solution 
of which reads 



P (%2 - 1)^/^^ ' ^^ 

"^^^"^"12 2/(1 - B) ^'3^ 

(H12 - B) 



From the H32 - H3_2 relation, ve had found for B: B = O.379. If, 

furthermore, we put, for adaptation to the test results, the number equal 
to 0.986, the function H-]_2(il) is, accordirig to figure 23, quite well 

satisfied by equation (l3)- 

Thus, the left sides of the energy equation (ll) and of the rela- 
tion B (equation (12)) seem to be identical. Then the right sides also 
must be equal which - solved with respect to b - results in 

Hip(A - 2e) + 2eB 
b = at) + -^ Cj'tI with A = I.269 



AHi2(Hio - 1) 



and B = 0.379 (l^) 



Therewith, a relation between qaantltiee of the velocity and of the 
shearing stress profile, based on the energy theorem, has been found for 
the assumption of the Gruechwitz method. In practice, of course, this 
equation is of no help, either, since no data concerning the newly intro- 
duced quantity e are available although one deals here, according to 
the defining equation (8), with a comparatively illustrative quantity. 
Thus one may conclude from equation (l4) only that it is improbable that 
invariably b = constant (as Gruschwitz assumes) or that b depends 
solely on U52/v (as Kehl presupposes), for b appears here as a func- 
tion of the respective velocity profile [r\ and Hx2) s-s well as of e 

and Cf' which likewise is not determined merely by U52/^. However, 

theoretically this conclusion is not cogent, either. It is, in itself, 
conceivable that the course of the t'orbulence mechanism is such that, 
in spite of different previous history, the complicated relations between 
T^, c^', and e have precisely the properties which cause b to be, 

for every rearward position, for instance a function of U62/V only and 
the test evaluation according to figure 2k actually shows that b, at 
least within a certain region, hardly varies. 



l6 NACA TM 131^ 



7. SUMMARY 



The report deals with measixrements in the turbxilent friction layer 
along a flat p]ate where the static pressiire from the leading edge of 
the plate onward systematically rises or decreases. In case of pressure 
rise, there results after a certain starting distance, a large increase 
in wall shearing stress to a m\iltiple of the initial value; this has 
already been briefly commented on in the report UM Nr. 6603. The further 
evaluation of the test material (calculations of the shearing stresses 
and mixing lengths) also gave qualitative information on this problem. 
As rule of thumb, it 'can merely be said that this strong increase in 

friction drag does not occur as -rr- -^^ < 2 X 10^ is valid. 
^ Q dx 

Furthermore, the empirical relations on which the Gruschwitz method 
is based are checked with the aid of the measurements. It is again 
confirmed that the tiirbulent velocity profiles form with sufficient 
accuracy a single -parameter family. Gruschwitz obtains from an empirical 
relation a differential equation for the variation of that parameter in 
flow direction; Kehl Improved that equation by not fixing a quantity b 
contained in it as constant, like Gruschwitz, but by considering it as 
a function of the Re-number of the friction layer. The present measure- 
ments confirm at first the simple relation b = constant. However, for 
higher Re-numbers of the friction layer - in the region of strongly 
increasing wall shearing stresses - different values for b resiilt, 
according to the previous history of the friction layer; here a relation 
of the form b = b(U62/v) is no longer sufficient, either. It is shown 

that this Gruschwitz -Kehl relation can be interpreted as statement of 
the energy theorem applied to friction layers. However, this energy 
theorem which simply signifies that the work of the t\irbvilent shearing 
stresses equals the loss of kinetic energy in the friction layer does 
not provide any practical help (for instance for setting up a calculation 
method) . As long as a sufficient statement for calculation of the 
shearing stresses themselves is lacking, a link between the newly intro- 
duced total work of the shearing stresses and the known friction layer 

quantities, -r^ displacement or momentum thickness, etc., also is 

lacking. For the same reason, it is not even possible to calculate the 
wall shearing stressj an approximation value for it must be inserted in 
the Gruschwitz method. Thus the result of the present investigation is. 



NACA TM 131^ 17 



on the whole, negative with respect to the problem of calciilating in 
advance turbulent friction layers; however, the test material represented 
in the figures might prove useful for f\irther theoretical considerations. 



Translated by Mary L. Mahler 
National Advisory Committee 
for Aeronautics 



18 NACA TM 131^ 

8. REFERENCES 



1. Gruschwitz, E.: Die tiu-bulente Reibungsschicht in ebener Stromung 

bei Druckabfall und Lruckanstieg. Ing. -Archiv, Bd. II, Heft 3, 
September 1931, PP- 321-3^6. 

2. Kehl, A. : Untersuchiingen iiber konvergente und divergente, turbulente 

Reibungsschichten. Ing.-Archiv, Bd. XIII, Heft 5, 19^3, pp. 293-329. 
(Available as R.T.P. Translation No. 2035, British Ministry of Air- 
craft Production. ) 

3. Walz, A. : Graphische Hilfsmittel zur Berechnung der laminaren und 

turbulenten Reibungsschicht. Lilientlial-Gesellschaft fur 
Luftfahrtforsch\ing Bericht S 10, 19^0, pp. h-^-lh. 

h. Walz, A.: Zur theoretischen Berechnung dea Hochstauftriebsbeivertes 
von Tragfluge±profilen ohne und mit Auftriebsklappen. ZWB For- 
scb.ungsbericht Nr. I769, March 19^3- 

5. Walz, A.: Naherungsverfahren z\ir Bei echr-'-^ng dei laminaren und 

turbulenten Reibungsschicht. Untersuchungen \ind Mitteilungen 
Nr. 3060. 

6. Wieghardt, K.: Ueber die Wandschubspannung in turbulenten Reibungs- 

schichten bei veranderlichem Aussendruck. Untersuchungen \ind 
Mitteilungen Nr. 6603. 

7. Wieghardt, K.: Ueber einen Energiesatz zur Berechnung laminarer 

Grenzschichten. (Available from CADO, Wright -Patterson Air Force 
Base, as ATI 33090.) 

8. Wieghardt, K.: Staurechen und Vielfachmanometer fur Messungen in 

Reibungsschichten. Technische Bericht, Bd. 11, No. 7, 19^4, p. 207. 

9- Prandtl, L.: Zur turbulenten Stromung in Rohren und langs Flatten. 
AVA-Ereebn. IV. Lief., 1932, p. I8. 

10. Scnultz-Grunow, F.: Neues Reibungswiderstandsgesetz fur glatte 

Flatten. Luftfahrtforschung, Vol. 17, No. 8, Aug. 20, 19^+0, 
p. 239- (Available as NACA TM 986.) 

11. Nikuradse, J.: Tvurbulente Reibiingsschichten. Publication of the 

ZWB, 19^2. 

12. Mangier, W.: Das Verhalten der Wandschubspannung in turbulenten 

ReibuTigsschichten mit Druckajistieg. Untersuchungen und Mitteilungen 
Nr. 3052, 19^3- 



NACA TM 131^^- 19 



13. Hartree, D. R. : On an Equation Occurring in FaUoier and Skan's 
Approximate Treatment of the Equations of the Boundary Layer. 
Proc. Cambridge Phil. Soc. , vol. 33, pt. 2, April 1937, PP- 223-239. 

ik. Buri, A.: Eine Berechnungsgrundlage fur die turhulente Grenzschicht 
bei beschleunigter und verzogerter Grundstromung. Eidgenossische 
Technische Hochschule, Zurich, 1931. (Available as R.T.P, 
Translation No. 2073, British Ministry of Aircraft Production.) 

15. Pretsch, J. : Zur theoretischen Berechnung des Profilviderstemdes. 
Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 60. (Avail- 
able as NACA TM 1009 •) 



20 



NACA TM I31I+ 



0.005 



0.004 
0.003 



0.002 



0.00 f 




0.006 
0.004 
0.003 
0.002 

0.001 



X 



U = const. = 17.8mA 



16 



2 



3 



U = const. = 33.0m/^ 




4 X [m] 5 



16 

12 



4 X [m] 5 



Figure 1,- Constant outer velocity, displacement, momentum, and energy 

thickness; local drag coefficient. 



NACA TM I31J+ 



21 




Figure 2.- Pressure rise: Case I. Displacement, momentum, and energy 

thickness; local drag coefficient. 



22 



NACA TM I31U 




Figure 3.- Pressure rise: Case n. Displacement, momentum, and energy 

thickness; local drag coefficient. 



NACA TM 1314 



23 







Figure 4,- Pressiire rise: Case HI. Displacement, momentum, and energy 

thickness; local drag coefficient. 



2k 



NACA ™ I31U 




Figure 5.- Pressure rise; Case IV. Displacement, momentum, and energy 

thickness; local drag coefficient. 



NACA OM 131^ 



25 



30 



\ 



20 



/ 



.+■ 



U 



y" 




^ 



•X. 



60 



p[kg/m^] 



-"^ 



40 







r 




Figure 6.- Pressure drop: Case V. Displacement, momentum, and energy 

thickness; local drag coefficient. 



26 



NACA TM I31I+ 




4 5 6 

Q dx 



Figure 7.- Cf' against — -^ for pressure rise and pressure drop; 
comparison with measurements of Kehl, 



NACA TM 131^^ 



27 








w 
w 

0) 

a, 

W 
W 

•rH 



cd 



2^H 



■o 



m 

o 

-i-i ^ 

-t-> 

:=) 

• r-t 

w 

•rH 

-o 

• rH 
O 
O 

1 — I 

0) 

> 

I 

00 

CD 



flH 



^^D 



28 



NACA TM 1311+ 




(U 

w 
w 

u 

a 

o 
a 

ctJ 



rt 






0) 

w 

O ■r< 

•1-1 M 
•4-' 

• rH 

KJ 

o 

r-H 
> 
I 

CTJ 



P4 



NACA TM 131^4- 



29 




U-- 1 7.6 mis 

• A = 0.4d7m 
+ 1.067m 

I.9d7m 
X 4.067m 



1.0 



0.8 



0.6 



0.4 



0.2 







-^ 


\x. 














\ 


X 




U -33.0 mis 

• X - 0.467m 
+ 1.087 m 
° 1.967m 
^ 4.067m 








N 


V 
























\ 


V, 


^^x. 


-4- •- 







2 



10 



" % '" 



Figure 10.- Dimensionless shearing stress profiles for constant outer velocity. 



30 



NACA TM I31I+ 




5 S ^ E E E 
t\ r\ i\ f. t\ rv. 

~~: c\i C\j o^i ri "O 



t~^ 



NACA TfL 131*+ 



31 




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o 

w 
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0) 
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W 

ho 

s 

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w 

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32 



NACA TM I31I+ 



^ ^ c\] .--j 05 t-j 'SJ ^ 



O <1 




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^ 




(U 




w 




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0) 


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a 

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NACA TM 131^^ 



33 




0) 

w 
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w 
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0) 

a 

w 
a; 



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0) 

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m 

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a 



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^ 



3^ 



NACA TM 13li^ 



Case U 




Case N 




Figure 15.- Shearing stress profiles and wall tangents (Buri form parameter). 



MCA TM 131^^ 



35 




0) 

w 
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0) 



1=1 

W 
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1 — I 

■rH 



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36 



NACA TM I31U 




0) 

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NACA TM'131^ 



37 




0) 

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38 



NACA TM I31I1 




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(in 



NACA TM 131^ 



39 







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a. 



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NACA TM 1314 




^ nmj ^ 



I 2 3 

Figure 21.- Variation of il for all measirring series. 




Figure 22.- Variation of H22 for all measuring series. (Significance of 
point markings, compare fig. 23.) 



NACA TM I31U 



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NACA TM 131^ 



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NACA TM 131^ 




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